I remember a story (true) from my friend Russ, in construction, and an avid geometer as well (with an eye for its art value -- not unlike George Hart and his daughter Vi).
This foundation they were building on seemed a bit off and when he asked the guy who laid it how he made sure it was rectangular, he said he'd measure the two diagonals, and they were equal. QED.
Of course if you think about it, an isosceles trapezoid has the same property, and that in fact is what they were building on, although the distortion was only apparent to the builders (they didn't start over as I recall).
Per my STEM standards (standards I'd apply versus those handed to me by authorities I do not recognize as such e.g. "state governors" (snicker)), the 1, 2, 3 power relationship between linear, areal and volumetric quantities would be harped on quite a bit.
Here's a simple computer game I sketched for this purpose:
This blog demo isn't precisely that issue, but what goes with it is the topic of "optimizing" and the fact that a sphere encloses the most volume for the same fixed amount of material.
Volume:Surface ratios would be a core topic. The fact this it's not a constant as a shape grows and shrinks (scales) is an important fact in nanotechnology as surface area becomes exponentially more compared to volumes enclosed (materials used, dose delivered).
I thought when I read the article first that we'd see kernel popcorn, not popped popcorn, going into the cylinders. That has the potential to get messier. I've often used beans for comparing volumes.
The beans I've used (e.g. in a Montessori demonstration -- then with students taking over) came with "mixing bowls" i.e. polyhedron containers open on one face so that beans might be poured from one to the other.
The same Russ (above) helped me build these things, out of stiff paperboard. http://www.flickr.com/photos/kirbyurner/479693617/
(when in primo condition)
Fill the tetrahedron with beans and pour into the octahedron 1, 2, 3, 4 times and it tops off. The edges are the same length. Now pour from the tetrahedron into the cube. Here, it's the face diagonals of the cube that match the tetrahedron's: 1, 2, 3 times and it tops off.
Yes, the ratios are exact, but given these are pre-schoolers and kindergarteners, we don't dive into the algebra right away.
You could call this a GeometryFirst [tm] curriculum in contrast to something more Gattegno (AlgebraFirst [tm] was more his trademark, what with the Cuisenaire Rods, color coded with matching letters for algebraic expressions).
This idea of using a tetrahedron as a unit of volume helps limber up the young brain and is a segue to other lesson plans I've already discussed in this thread, such as this one: